Abstract
We present a method that generates two-sided sequent calculi for four-valued logics like first degree entailment (FDE). (We say that a logic is FDE-like if it has finitely many operators of finite arity, including negation, and if all of its operators are truth-functional over the four truth-values ‘none’, ‘false’, ‘true’, and ‘both’, where ‘true’ and ‘both’ are designated.) First, we show that for every n-ary operator ⋆ every truth table entry f⋆(x1,…,xn) = y can be characterized in terms of a pair of sequent rules. Secondly, we use these sequent rules to build sequent calculi and prove their completeness. With the help of two simplification procedures we then show that the 2 ⋅ 4n sequent rules that characterize an n-ary operator can be systematically reduced to at most four sequent rules. Thirdly, we use our method to investigate the proof-theoretical consequences of including intuitive truth-functional implications in FDE-like logics.
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Acknowledgements
The authors thank Chris Fermüller, Rosalie Iemhoff, João Marcos, Sara Negri, Yuri David Santos, Yaroslav Shramko, Rineke Verbrugge, Heinrich Wansing, the reviewer of this journal, and the audiences who attended our presentations at the University of Amsterdam, the University of Greifswald, the Universidade da Madeira, and the École normale supérieure (Paris) for their questions, comments, and suggestions.
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Kooi, B., Tamminga, A. Two-sided Sequent Calculi for FDE-like Four-valued Logics. J Philos Logic 52, 495–518 (2023). https://doi.org/10.1007/s10992-022-09678-0
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DOI: https://doi.org/10.1007/s10992-022-09678-0